Creating Deltahedra with Unfolded Net of Tetrahedron

Year: 2019 Authors: Liang-Chen Peng

Core claim

A tetrahedron net can be redesigned as a modular unit for constructing many deltahedra and related polyhedra in classroom settings.

Topics

deltahedra, origami-inspired modular construction, geometry education, polyhedral models

Domains

polyhedra, tetrahedron nets, octahedron and icosahedron, Archimedean solids, origami, cardboard craft, geometry-based artwork, decorative tiling patterns

Methods

modular folding, interlocking tabs and pockets, classroom model building, iterative design refinement

Media

cardboard, tabs and pockets, glue, double-sided tape

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2019 Conference Proceedings

Creating Deltahedra with Unfolded Net of Tetrahedron

Liang-Chen Peng

The Affiliated Senior High School of National Taiwan Normal University, Taipei, Taiwan 694@hs.ntnu.edu.tw

Abstract

This paper presents a new origami-inspired design to create polyhedra. By using the unfolded net of tetrahedron as modular units, one can assemble a variety of deltahedra. This design was used to engage students in geometry classes by having them build polyhedral models from specially designed cardboards. Through this exercise, students learned about the alternative features of polyhedra in a hands-on manner and appreciated the dynamic interaction among math, origami, and art.

Introduction

Origami designers typically need to pre-fold many identical modular units [4,5,6], a very time-consuming process unsuitable for classroom use. I developed special cardboard units for students to efficiently make deltahedra [1,8], incorporating origami techniques. This design is intended to encourage students to think spatially while making geometry-based artwork. It was used to promote the Math-Maker course of The Affiliated Senior High School of National Taiwan Normal University. Students learned the relationship within polyhedral structures by being makers themselves. The exhibition of student’s works had the slogan of “Math, Origami, and Art.”

Basic Design

Figure 1a, a simple origami-inspired design, shows the initial step of using such modular units. One can create bipyramid, octahedron, and some elevated polyhedra [7], by assembling these basic units, as shown in Fig. 1b.

Figure 1: Unit diagrams (a) and the works of deltahedra (b).

Improved Design

For regular tetrahedron and icosahedron, the mirror-symmetrical part unit from the basic design above is used. I modified the tabs and pockets of Fig. 2b so it can be combined with Fig. 2a to form a new unit, Fig. 2c. Since Fig. 2c is the zipper unfolding of tetrahedron [3], and the tabs and pockets are arranged to interlock, according to the zipper principle of unfolded polyhedron [2], it is now easy to complete the models, following the equal segmentation of the unfolded nets and the interlocking tabs and pockets. Fig. 2d shows the net of octahedron combined with two units; Fig. 2e shows the net of icosahedron combined with five units.

Peng

Figure 2: The nets of octahedron (d) & icosahedron (e) made up with the nets of tetrahedron (c).

Assembling an elevated polyhedron with these units requires more spatial thinking. One strategy is to first fold the units into Fig. 3a and its mirror-symmetrical counter-part Fig. 3b, and then further combine them into an elevated tetrahedron (Fig. 3c) and an elevated octahedron (Fig. 3d), similar to the common way of combining origami units with the interlocking tabs and pockets. If the final assembled structure is loose, the cardboard is over-stressed by the valley-folded structure. In this case, glue or double-sided tapes can be used to fix the problem.

Figure 3: Assemble elevated polyhedron based on unit origami.

Even more challenging is the elevated icosahedron. One can first make up a “southern hemisphere” (Fig. 4b) and a “northern hemisphere” (Fig. 4a) with 5 units (Fig. 3a) and 5 mirror-symmetrical units (Fig. 3b), respectively. Then the “equator” (Fig. 4c) can be created with 5 unfolded units. What remains is to connect the two hemispheres with the equator with the interlocking tabs and pockets.

Creating Deltahedra with Unfolded Net of Tetrahedron

Figure 4: Assemble elevated icosahedron based on a globe.

Improvement

After several iterations of design, the unit is more multifunctional. I added more fold lines to the schematic diagram to create two types of tiling units, which bear consistent mirror symmetry with each other for most economical use of the layout. One unit can be folded at the trisection of edges (Fig. 5a); the other can be folded at the bisection of edges (Fig. 5b). With these multifunctional units, it is now possible to construct variations of Archimedean solids, with craters instead of “vertices” (Fig. 5c). The template of the final design, used for mass production, with Sierpinski triangle or Koch snowflake as decoration, is attached below (Fig. 6).

Figure 5: Multifunctional units $(a,b)$ for more polyhedra.

Conclusion

This paper explained how to use multifunctional units to construct various types of deltahedra. The process of folding and combining the units into deltahedral structures engages the learners to think spatially while they make satisfying art with their own hands.

Peng

Acknowledgements

I wish to thank the Ministry of Science and Technology, Taiwan, R. O. C. High-Scope Project No. MOST107-2514-S-788-002 for their financial support. I am also grateful for Tsan-Kuang Lee’s editing help.

References

[1] Chia-Chin Tsoo and Bih-Yaw Jin. “Designing Skeletal Polyhedral Sculptures Inspired by Octet-Truss Systems and Structural Inorganic Chemistry with Bugle Beads.” Bridges Conference Proceedings, Waterloo, Ontario, Canada, July. 27–31, 2017, pp. 483–486. http://archive.bridgesmathart.org/2017/bridges2017-483.. [2] Chun-Chih Peng. “Something About the Unfolded Nets of Polyhedra–Rotational Symmetry & Zipper Principle.” Science Education Monthly, vol.234, no. 11, 2000, pp. 11–18. http://www.sec.ntnu.edu.tw/monthly/89(231-240)/234pdf/%E5%B1%95%E9%96%8B%E5%9C%96%E4%BA%8C%E4%B8%89%E4%BA%8B.pdf. [3] Demaine, Erik D.; Demaine, Martin L.; Lubiw, Anna; Shallit, Arlo; Shallit, Jonah L. “Zipper Unfoldings of Polyhedral Complexes.” Canadian Conference on Computational Geometry 2010, Winnipeg MB, Aug. 9-11, 2010. https://erikdemaine.org/papers/Zippers_CCG2010/short.pdf [4] Heidi Burgiel. “Unit Origami: Star-Building on Deltahedra.” Bridges Conference Proceedings, Baltimore, Maryland, USA, July. 29-Aug. 1, 2015, pp. 585-588. http://archive.bridgesmathart.org/2015/bridges2015-585.. [5] Liang-Chen Peng. “Single, Double, Triangle: The Math-Magic from Triangle to Deltahedra.” Science Study Monthly, vol.56, no. 5, 2017, pp. 34-41. https://activity.ntsec.gov.tw/activity/ssm/56_5/HTML/mobile/index.#p=34. [6] Nick Fout and Jenn Marker. “Folding Your Way to Understanding.” Bridges Conference Proceedings, Coimbra, Portugal, July. 27–31, 2011, pp. 671–678. http://archive.bridgesmathart.org/2011/bridges2011-671.. [7] Rinus Roelofs. “The Elevation of Coxeter’s Infinite Regular Polyhedron 444444.” Bridges Conference Proceedings, Jyväskylä, Finland, Aug. 9–13, 2016, pp. 33–40. http://archive.bridgesmathart.org/2016/bridges2016-33.. [8] Weiling He, Chengde Wu and Ergun Akleman. “Constructing Deltahedra from Recycled Plastic Bottles.” Bridges Conference Proceedings, Waterloo, Ontario, Canada, July. 27–31, 2017, pp. 459–462. http://archive.bridgesmathart.org/2017/bridges2017-459..

Attachments

Figure 6: The final design cardboard with Sierpinski triangle or Koch snowflake as decoration.